The one-dimensional Keller-Segel model with fractional diffusion of cells

We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent $0<\alpha\leq 2$. We prove some features related to the classical two-dimensional Keller-Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when $\alpha<1$ and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for $\alpha\leq1$ if the initial density is small enough in the sense of the $L^{1/\alpha}$ norm.Comment: 12 page

Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all the solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension 2, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d?3$

Mathematical description of bacterial traveling pulses

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on {\em E. coli} have shown precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at a macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This model can account for recent experimental observations with {\em E. coli}. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition we can capture quantitatively the main characteristics of the pulse such as the speed and the relative size of tails. This work opens several experimental and theoretical perspectives. Coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance the stiffness of the signal integration process turns out to have a strong effect on collective motion. Furthermore the bottom-up scaling allows to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion

From microscopic to macroscopic descriptions of cell\ud migration on growing domains

Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs

Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact

Clinical outcome and prognostic factors for patients treated within the context of a phase I study: the Royal Marsden Hospital experience

The main aim of phase I trials is to evaluate the tolerability and pharmacology of a new compound. However, investigating the potential for clinical benefit is also a key objective. Our phase I trial portfolio incorporates a range of new drugs, including molecular targeted agents, sometimes given together with cytotoxic agents. We performed this analysis of response rate, progression-free (PFS) and overall survival (OS) to assess the extent of clinical benefit rate (CBR: partial response (PR)+stable disease (SD)) derived from current trials. We analysed 212 consecutive patients who were treated in 29 phase I studies, from January 2005 to June 2006. All patients had progression of disease prior to study entry. The median age was 58 years (range: 18–86) with a male/female ratio of 2 : 1. A total of 148 patients (70%) were treated in ‘first in human trials' involving biological agents (132 patients) or new cytotoxic compounds (16 patients) alone, and 64 patients (30%) received chemotherapy-based regimens with or without biological agents. After a median follow-up time of 34 weeks, the median PFS and OS were 11 and 43 weeks, respectively. The CBR was 53% (9% PR and 44% SD) after the first tumour evaluation after two cycles (between weeks 6 and 8) and has been maintained at 36 and 26% at 3 and 6 months, respectively. Treatment related deaths occurred in 0.47% of our patients and treatment had to be withdrawn in 11.8% of patients due to toxicity. A multivariate analysis (MVA) of 13 factors indicated that low albumin (<35 g l−1), lactate dehydrogenase>upper normal limit and >2 sites of metastasis were independent negative prognostic factors for OS. A risk score based on the MVA revealed that patients with a score of 2–3 had a significantly shorter OS compared to patients with a score of 0–1 (24.9 weeks, 95% CI 19.5–30.2 vs 74.1 weeks, 95% CI 53.2–96.2). This analysis shows that a significant number of patients who develop disease progression while receiving standard therapy derived benefit from participation in phase I trials. Risk scoring based on objective clinical parameters indicated that patients with a high score had a significantly shorter OS, and this may help in the process of patient selection for phase I trial entry

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated

Borrelia recurrentis employs a novel multifunctional surface protein with anti-complement, anti-opsonic and invasive potential to escape innate immunity

Borrelia recurrentis, the etiologic agent of louse-borne relapsing fever in humans, has evolved strategies, including antigenic variation, to evade immune defence, thereby causing severe diseases with high mortality rates. Here we identify for the first time a multifunctional surface lipoprotein of B. recurrentis, termed HcpA, and demonstrate that it binds human complement regulators, Factor H, CFHR-1, and simultaneously, the host protease plasminogen. Cell surface bound factor H was found to retain its activity and to confer resistance to complement attack. Moreover, ectopic expression of HcpA in a B. burgdorferi B313 strain, deficient in Factor H binding proteins, protected the transformed spirochetes from complement-mediated killing. Furthermore, HcpA-bound plasminogen/plasmin endows B. recurrentis with the potential to resist opsonization and to degrade extracellular matrix components. Together, the present study underscores the high virulence potential of B. recurrentis. The elucidation of the molecular basis underlying the versatile strategies of B. recurrentis to escape innate immunity and to persist in human tissues, including the brain, may help to understand the pathological processes underlying louse-borne relapsing fever

Rechargeable Batteries of the Future—The State of the Art from a BATTERY 2030+ Perspective

The development of new batteries has historically been achieved through discovery and development cycles based on the intuition of the researcher, followed by experimental trial and error—often helped along by serendipitous breakthroughs. Meanwhile, it is evident that new strategies are needed to master the ever-growing complexity in the development of battery systems, and to fast-track the transfer of findings from the laboratory into commercially viable products. This review gives an overview over the future needs and the current state-of-the art of five research pillars of the European Large-Scale Research Initiative BATTERY 2030+, namely 1) Battery Interface Genome in combination with a Materials Acceleration Platform (BIG-MAP), progress toward the development of 2) self-healing battery materials, and methods for operando, 3) sensing to monitor battery health. These subjects are complemented by an overview over current and up-coming strategies to optimize 4) manufacturability of batteries and efforts toward development of a circular battery economy through implementation of 5) recyclability aspects in the design of the battery

Export-Oriented International Joint Venture: Endogenous Set-Up Costs and Information Gathering

We analyze the formation of an export-oriented international joint venture (IJV) between a multinational corporation (MNC) and a domestic firm under demand uncertainty and in a principal-agent framework. The MNC possesses a superior production technology and is better at predicting foreign market demand. The domestic firm can reduce set-up costs of the IJV with effort levels that is endogenously determined. We examine how the MNC\u27s preference for, and the ownership structure of, a joint venture depend on the efficiency of information gathering and of cost reduction, and on the nature of credit markets. We find, inter alia, that when the credit constraint is severe the MNC does not push the domestic firm to its reservation profit level. A relaxation of the credit constraint facing the domestic firm never makes it better off and in fact makes the domestic firm worse off when the credit constraint is severe